). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become

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1 Relativity and quantum mehanis: Jorgensen 1 revisited 1. Introdution Bernhard Rothenstein, Politehnia University of Timisoara, Physis Department, Timisoara, Romania. brothenstein@gmail.om Abstrat. We first define the funtions whih ensure the transformation of momentum and energy of a tardyon, the transformation of the wave vetor and the frequeny of the assoiated wave. Having done this, we show that they ensure the relativisti invariane of the quotient between momentum and wave vetor and between energy and frequeny if the produt between partile veloity u and phase veloity w is a relativisti invariant (uw= ), a ondition whih is a natural ombination of speial relativity theory and quantum mehanis. Jorgensen 1 starts his paper entitled Relativity and quantum by stating that beams of entities, suh as eletrons, may produe diffration patterns. These patterns may be interpreted in terms of partiles and waves. One obvious question onerning this phenomenon is what is the funtional relation between the momentum of the entity and its momentum? While this relation is well known, it is of interest to look for another way to arrive at this funtion by using speial relativity theory and the fundamental observation that the mathematial form of a law of nature annot ontain any parameters relating to more than one referene frame. The purpose of our paper is to analyze the properties involved in the transformation of the spae-time oordinates of the same event, of the mass (energy) and momentum of the same tardyon and of the frequeny and the wave vetor of a wave propagating at subluminal veloities; this last enables us to introdue the onept of phase veloity.. What do transformation equations share in ommon? Consider an eletron that moves with veloity u relative to the inertial referene frame I along a diretion that makes an angle θ with the positive diretion of the OX axis. After a given time of propagation t the eletron arrives at a point M ( x = r os! = ut os!; y = r sin! = ut sin! ) generating the event E( x = r os! = ut os!; y = r sin! = ut sin!; t = r / u = x + y / u), using Cartesian (x,y) and polar (r,θ) oordinates in a two spae dimensions approah. Deteted from the I inertial referene frame that moves with onstant veloity relative to I in the positive diretion of the overlapped OX(O X ) axes, the same event is E" ( x" = r" os! " = u" t" os!", t" = r" / u" = x" + y" / u" ). In aordane with the Lorentz transformations for the spae-time oordinates of the same event, the spae oordinates beome 1

2 1/ # $ r% =! r ( os " & / u) + (1 & / )sin " =! r' 1( u,,, " ) (1) ( ) u t% # $ =! t ( 1& os " t ) =! ' ( u,,", ). () * + We underline an important property of the funtions defined above: both of them have the same limit for u!, i.e. # 1, = lim # 1( u,,!, ) = 1$ os! (3) u" and $, = lim $ ( U,,!, ) = 1% os! (4) x"#.. Transformation equations for the mass, energy and momentum of the same eletron Let m, E and p be the mass, energy and momentum of an eletron when deteted from I and m, E and p when deteted from I. By definition p = mu (5) p! = m! u! (6) as long as we are interested in the magnitudes of the physial quantities involved. Combining (5) and (6), we obtain p" p u" p # 1( u,,!, ) = =. (7) m" m u m # ( U,,!, ) Equation (7) suggests onsidering that p" = F( )# 1( u,,!, ) (8) and m" = F( )# ( u,,!, ) (9) where F() represents an unknown funtion whih depends only on the relative veloity of the inertial referene frames involved. We obtain this funtion by imposing u=0 in (9). Under suh onditions, observers from I measure the rest mass of the eletron as m 0, whereas observers from I measure its inertial mass m given by m = F( ) m0!, u= 0 = F( ) m0. (10) Experiment and speial relativity ombined with onservation laws of momentum and energy 3 leads to the following relationship between rest mass and inertial mass of the same eletron m =! m 0 (11) with the result that 1 F( ) =! =. (1) 1"

3 Equations (8) and (9) beome p# =! p$ ( u,,", ) (13) 1 m# =! m$ ( u,,", ). (14) Multiplying both sides with, we obtain that energy transforms as E# =! E$ ( u,,", ). (15) Jorgensen 1 proposes a transparent riterion for finding out if a ombination of physial quantities is a relativisti invariant or not. Mimiking his approah we start, with p! p 1 $ / u p " % u 1 &% u & = = 1 + $ # m! ' (' ( m 1 $ u / m ) u * (16) - +,+,. in the ase of a single spae dimension (θ=θ =0). We see that p p! = only in m m! the ase of a photon (u=u =), in whih ase the quotient between the momentum and the inertial mass of the photon is the invariant..3. Transformation equation for frequeny and wave vetor Moller 3 derives transformation equations for the parameters introdued in order to haraterize a plane wave propagating with subluminal veloity. The parameters mentioned are unit vetor n that haraterizes the diretion in whih the wave propagates, the phase veloity f w, the frequeny f, the wave vetor k = n, and the wavelength λ when w f! deteted from I, but n,w,f, k! = n! and!" when deteted from I. Using w! results obtained by this author and extending them to the ase of the wave vetor, we have the following transformation equations f % # $ =! f ( 1& os " ) =!' 1(, w, " ) (17) * w + for the frequenies of the osillations taking plae in the wave and w # w $ k% =! k & " + & " =! k' u " * + 1 os ( sin ) (,,, ) (18) for the magnitude of the wave vetor. The funtions ψ 1 and ψ defined above have the same limit for w= equal to $ % w $ w % & = lim ( 1' os! ) = lim 1' os! + sin! 1 os! x"# x ( ' ) = ' (19) w "# * + * + 3. Quantum mehanis and speial relativity 3

4 De Broglie 4 assoiates a wave that propagates with phase veloity w to a tardyon moving with veloity u, establishing the following relationships between momentum, wave vetor, energy and frequeny p h k = (0) E h! = (1) where! and h stand for the frequeny of the osillations in the wave and for Plank s onstant, respetively. Beause universal onstants are relativisti invariants, we expet that the ombinations of physial quantities P/k and E/! should have the same magnitude in all inertial referene frames in relative motion. As we an see, De Broglie s equations (0) and (1) ombine a physial quantity whih haraterizes the partile property (P,E) and a physial quantity that haraterizes the wave property (k,! ). Taking into aount the results obtained above, we an present (0) and (1) as " # " # % os! $ & + % 1$ sin &! * 1( u,,!, ) u ( ) = = (,,!, ) 1$ os! + $ sin! p' p p ( ) () k' k + w k w " w # % & ( ) u 1 os E$ #! E % ( u,,!, ) = =. (3) " $ " & 1(, w,! ) 1# os! w Analyzing the properties of the funtions involved in the transformation proess, we see that all of them have the same limit, not only for u=w=, but also for uw Moller 3 derives ondition (4) by imposing the ondition that the partile moves and the wave propagates along the same diretion. A ombination of quantum mehanis and speial relativity leads to the same result and goes as follows. Consider that, in aordane with De Broglie, we have =. (4) m0u k = u h 1! and using the relativisti energy m0 E = = h! u 1" (5) (6) 4

5 where m 0 stands for the rest mass of the partile, ombining (5) and (6), we obtain uw = (7) i.e. the ondition whih ensures the invariane of p/k and of E/!, and demonstrates the intimate relationship between speial relativity theory and quantum mehanis. Aknowledgment I wish to aknowledge the help of A. Pearlstein in the elaboration of the present paper who guided me through Jorgensen s paper. Referenes 1 Theodore P. Jorgensen, Relativity and the quantum, International Journal of Theoretial Physis, 37, (1998) Yuan Zhong Zhang, Speial Relativity and its Experimental Foundations, (World Sientifi 1996) pp C.Moller, The Theory of Relativity, (Clarendon Press Oxford, 197) Ch.3 4 Jay Orear, Physik, (Carl Hanser erlag Munhen, Wien 1979) pp